Based on work of Caffarelli and Silvestre [1]}, Stinga and Torrea proposed in [2]} to define \(L^\sigma f\)  as the
Neumann boundary value associated with the elliptic PDE
\(\left\lbrace \begin{array}{lll} u_{tt} +\tfrac{1-2\sigma }{t}\,u_t\,=\,L u, & & t>0\\u(0,x)=f(x).&&\\\end{array}\right.\) 
